200
L23
§5.3 Logarithmic Functions
The exponential function
( )
x
f x
a
=
(
0,
1)
a
a
>
≠
is a
1-1 function and, therefore, it has the inverse
1
( )
f
x
−
.
We denote the inverse log
a
x
and read “logarithm to
the base
a
of
x
”.
Sketch the graphs of
2
x
y
=
and
2
log
y
x
=
.
Sketch the graphs of
1
2
x
y
⎛
⎞
=
⎜
⎟
⎝
⎠
and
1
2
log
y
x
=
201
We use what we know about
x
y
a
=
and inverses to get
the properties of
log
a
y
x
=
.
( )
x
f x
a
=
1
( )
log
a
f
x
x
−
=
Points (0,1) and (1,
)
a
are on the graph
Points (
,
) and (
,
) are
on the graph
0
y
=
is HA
Domain: (
,
)
−∞ +∞
Domain: (
,
)
Range: (0,
)
+∞
Range: (
,
)
log
x
a
a
x
=
for any real
x
log
a
x
a
x
=
for
0
x
>
**Important**
Since inverse functions undo each other with respect to
composition, the following
identities
hold:
log
for all real
x
a
a
x
x
=
log
for
0
a
x
a
x
x
=
>
.
We use these identities for solving exponential and
logarithmic equations and inequalities.

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